src/ZF/arith.thy
 author lcp Tue, 05 Oct 1993 15:21:29 +0100 changeset 25 3ac1c0c0016e parent 6 8ce8c4d13d4d child 124 858ab9a9b047 permissions -rw-r--r--
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many changes epsilon,arith: many changes ordinal/succ_mem_succI/E: deleted; use succ_leI/E nat/nat_0_in_succ: deleted; use nat_0_le univ/Vset_rankI: deleted; use VsetI
```
(*  Title: 	ZF/arith.thy
ID:         \$Id\$
Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory

Arithmetic operators and their definitions
*)

Arith = Epsilon +
consts
rec  :: "[i, i, [i,i]=>i]=>i"
"#*" :: "[i,i]=>i"      		(infixl 70)
div  :: "[i,i]=>i"      		(infixl 70)
mod  :: "[i,i]=>i"      		(infixl 70)
"#+" :: "[i,i]=>i"      		(infixl 65)
"#-" :: "[i,i]=>i"      		(infixl 65)

rules
rec_def  "rec(k,a,b) ==  transrec(k, %n f. nat_case(a, %m. b(m, f`m), n))"

add_def  "m#+n == rec(m, n, %u v.succ(v))"
diff_def "m#-n == rec(n, m, %u v. rec(v, 0, %x y.x))"
mult_def "m#*n == rec(m, 0, %u v. n #+ v)"
mod_def  "m mod n == transrec(m, %j f. if(j<n, j, f`(j#-n)))"
div_def  "m div n == transrec(m, %j f. if(j<n, 0, succ(f`(j#-n))))"
end
```